We see here in this case decoherence is completely affected by \(\lambda _{k,p}\equiv \frac{C_{p}(\tau )}{C_{k}(\tau )}\) . In fact if \(\lambda \)  is very small then decoherence happens weakly but if this is very large then decoherence happens rapidly based on how the behavior of \(\mathcal {H}^{(1)}\)  is.
There are two important key points: first, the last term in the right hand side Eq. (REF ) has no effect on decoherence because the imaginary part of \(F\)  plays role in the decoherence; second, in contrast to the Eq. (REF ) which the imaginary part of \(F\)  is proportional to the integration of \( \mathcal {H}^{(2)} \) , the imaginary part of \(F\)  is related to the integration of \( \mathcal {H}^{(1)} \) . As a result, \( \mathcal {H}^{(1)} \)  has to satisfy the same condition needed for \( \mathcal {H}^{(2)} \)  to have decoherence [1]}, [2]}.
